Nuprl Lemma : uniform-continuity-from-fan2
∀[T:Type]
((∃U:Type. ((U ⊆r ℕ) ∧ (∃r:ℕ ⟶ U. ∀x:U. ((r x) = x ∈ U)) ∧ (∃h:T ⟶ U. Bij(T;U;h))))
⇒ (∀F:(ℕ ⟶ 𝔹) ⟶ T. ⇃(∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕn ⟶ 𝔹)) ⇒ ((F f) = (F g) ∈ T)))))
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
isl: isl(x),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
guard: {T}
Lemmas referenced :
uniform-continuity-from-fan,
strong-continuity2-half-squash-surject-biject,
bool_wf,
surject-nat-bool,
biject_wf,
implies-quotient-true,
nat_wf,
int_seg_wf,
unit_wf2,
equal_wf,
subtype_rel_function,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
assert_wf,
btrue_wf,
bfalse_wf,
sq_exists_wf,
istype-nat,
istype-assert,
subtype_rel_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
productElimination,
hypothesis,
Error :dependent_pairFormation_alt,
sqequalRule,
Error :functionIsType,
Error :universeIsType,
Error :equalityIstype,
because_Cache,
applyEquality,
productEquality,
functionEquality,
natural_numberEquality,
setElimination,
rename,
unionEquality,
independent_isectElimination,
independent_pairFormation,
Error :inlEquality_alt,
isectEquality,
Error :inhabitedIsType,
unionElimination,
equalityTransitivity,
equalitySymmetry,
Error :lambdaEquality_alt,
Error :unionIsType,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
Error :isectIsType,
instantiate,
universeEquality
Latex:
\mforall{}[T:Type]
((\mexists{}U:Type. ((U \msubseteq{}r \mBbbN{}) \mwedge{} (\mexists{}r:\mBbbN{} {}\mrightarrow{} U. \mforall{}x:U. ((r x) = x)) \mwedge{} (\mexists{}h:T {}\mrightarrow{} U. Bij(T;U;h))))
{}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} T. \00D9(\mexists{}n:\mBbbN{}. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((f = g) {}\mRightarrow{} ((F f) = (F g))))))
Date html generated:
2019_06_20-PM-02_52_26
Last ObjectModification:
2019_02_06-PM-04_01_15
Theory : continuity
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